Sufficiently complex math is able to synthesize speech and draw pictures. Yes, complex math is able to predict the behavior of the quantum world. So what? OpenGL is also able to predict realistic pictures. Is knowledge in OpenGL mechanics will improve our knowledge in the physics?
My teacher told me, that I will really know physics when I will be able to derive my own equations. I extended this definition to "I will really know physics when I will be able to create the physical model of a physical thing, to be able to watch it with my own eyes".
Agreed, to an extent. But some of the parts of QM seem like things reality demands (double slit experiments) and other parts seem like horrible mathematical artifacts that in enough years we'll come to regret (like the Schrödinger equation). But of course that's not a stance I can defend unless we figure out a better mathematical model for it. Just a hunch.
Yes, but the reason you're learning those things is that that is the way reality actually works. That's what happens when you actually do experiments. Richard Feynman once remarked: "quantum mechanics was not wished upon us by theorists". We have QM because physicists were dragged to it kicking and screaming by actual experiments that showed them that reality does not work the way classical physics describes. And the reason all those difficult mathematical tools have to be taught is that that's the only way physicists know how to build models that make correct predictions about what happens when you do those experiments.
In other words, yes, you can say QM doesn't make sense, but that's because reality doesn't make sense--at least not if "make sense" means "works the way my human intuitions, which never evolved to deal with quantum phenomena, say it should". Yes, reality doesn't work the way our human intuitions say it should. That means we have to retrain our intuitions--which is what a lot of learning QM is really about.